Stress relaxation modulus definition

The stresses in the two experiments at the same time t are proportional to the imposed strains. Tliis fact leads to the definition of the stress-relaxation modulus at time t ... 

In real food polymers, a distinction can be made between a viscoelastic solid, which contains some cross-links that are permanent, and a viscoelastic liquid, where, under the influence of stress, the relative movement of whole molecules will be observed. As shown in Figure 8.9, in the case of a viscoelastic solid, after application of the stress, the strain will eventually reach a constant value, and upon removal of the stress, the strain will finally return to the remaining value of food primary energy, which was not entirely dissipated. For a viscoelastic liquid, a permanent deformation will remain after removal of the stress. In the stress relaxation area, the deformation value will decay to zero for a viscoelastic liquid, whereas for a solid, it will reach a constant, nonzero value. It can also be seen as either a decreased value to the zero or a constant, nonzero value, as pointed out by the dashed line. Both values characterize the rheology parameters of foods under certain conditions. One of the reasons for this is that the factors of time-dependent foods are not necessarily related to their elastic modulus. This can be explained by the series of small deformations without rupture, which are dependent in different ways and are based on the primary molecular microstructure of foods. The time required for the stress to relax to a definite fraction of its initial value is the relaxation time. 

Equation 5.18 involves the experimentally observed plateau modulus, and this assumes that this quantity reflects aU the relaxation that occurs in response to the initial stress, except for the extremely short-time glassy modes. However, relatively fast Rouse modes of relaxation allow re-equilibration of tension along the chain, and as a result, one fifth of the initial stress relaxes before the entanglement network interrupts the process. Thus, the plateau modulus actually observed in an experiment is expected to be about 4/5 of the quantity on the right in Eq. 5.18. This suggests the alternative definition of Mg for entangled melts shown below. 

Service temperatures are definitely lower under mechanical stress because of modulus decay, strain, creep, relaxation. .. They can be of the order of 50°C up to 80°C according to the HDT and applied stresses. 

There is a single dimensionless group, XVjL, which is known as the Weissenberg number, denoted by various authors as We or Wi. (We is more common, but it can lead to confusion with the Weber number, so Wi will be used here.) The shear rate in any viscometric flow is equal to a constant multiplied by V/L, so it readily follows that the ratio of the first normal stress difference to the shear stress is equal to twice that constant multiphed by Wi. Hence, Wi can be interpreted as the relative magnitude of elastic (normal) stresses to shear stresses in a viscometric flow. The ratio of the shear stress to the shear modulus, G, is sometimes known as the recoverable shear and is denoted Sr. Sr differs from Wi for a Maxwell fluid only by the constant that multiplies F jL to form the shear rate for a given flow. In fact, many authors define Wi as the product of the relaxation time and the shear rate, in which case Wi = Sr. It is important to keep the various definitions of Wi in mind when comparing results from different authors. 

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